Information Retrieval and Topic Models. Mausam (Based on slides of W. Arms, Dan Jurafsky, Thomas Hofmann, Ata Kaban, Chris Manning, Melanie Martin)

Transcription

1 Information Retrieval and Topic Models Mausam (Based on slides of W. Arms, Dan Jurafsky, Thomas Hofmann, Ata Kaban, Chris Manning, Melanie Martin)

2 Sec. 1.1 Unstructured data in 1620 Which plays of Shakespeare contain the words Brutus AND Caesar but NOT Calpurnia? One could grep all of Shakespeare s plays for Brutus and Caesar, then strip out lines containing Calpurnia? Why is that not the answer? Slow (for large corpora) NOT Calpurnia is non-trivial Other operations (e.g., find the word Romans near countrymen) not feasible Ranked retrieval (best documents to return) Later lectures 2

3 Sec. 1.1 Term-document incidence matrices Antony and Cleopatra Julius Caesar The Tempest Hamlet Othello Macbeth Antony Brutus Caesar Calpurnia Cleopatra mercy worser Brutus AND Caesar BUT NOT Calpurnia 1 if play contains word, 0 otherwise

4 Sec. 1.1 Incidence vectors So we have a 0/1 vector for each term. To answer query: take the vectors for Brutus, Caesar and Calpurnia (complemented) bitwise AND AND AND = Antony and Cleopatra Julius Caesar The Tempest Hamlet Othello Macbeth Antony Brutus Caesar Calpurnia Cleopatra mercy worser

5 Sec. 1.1 Answers to query Antony and Cleopatra, Act III, Scene ii Agrippa [Aside to DOMITIUS ENOBARBUS]: Why, Enobarbus, When Antony found Julius Caesar dead, He cried almost to roaring; and he wept When at Philippi he found Brutus slain. Hamlet, Act III, Scene ii Lord Polonius: I did enact Julius Caesar I was killed i the Capitol; Brutus killed me. 5

6 Sec. 6.2 Term-document count matrices Consider the number of occurrences of a term in a document: Each document is a count vector in N V : a column below Antony and Cleopatra Julius Caesar The Tempest Hamlet Othello Macbeth Antony Brutus Caesar Calpurnia Cleopatra mercy worser

7 Sec tf-idf weighting The tf-idf weight of a term is the product of its tf weight and its idf weight. w (1 log tf ) log t, d 10 ( N Best known weighting scheme in information retrieval Note: the - in tf-idf is a hyphen, not a minus sign! Alternative names: tf.idf, tf x idf Increases with the number of occurrences within a document Increases with the rarity of the term in the collection / df t, d t )

8 Sec Final ranking of documents for a query Score(q,d) = å t ÎqÇd tf.idf t,d 9

9 Sec. 6.3 Binary count weight matrix Antony and Cleopatra Julius Caesar The Tempest Hamlet Othello Macbeth Antony Brutus Caesar Calpurnia Cleopatra mercy worser Each document is now represented by a real-valued vector of tf-idf weights R V

10 Sec. 6.3 Documents as vectors Now we have a V -dimensional vector space Terms are axes of the space Documents are points or vectors in this space Very high-dimensional: tens of millions of dimensions when you apply this to a web search engine These are very sparse vectors most entries are zero

11 Sec. 6.3 Queries as vectors Key idea 1: Do the same for queries: represent them as vectors in the space Key idea 2: Rank documents according to their proximity to the query in this space proximity = similarity of vectors proximity inverse of distance Recall: We do this because we want to get away from the you re-either-in-or-out Boolean model Instead: rank more relevant documents higher than less relevant documents

12 Sec. 6.3 Formalizing vector space proximity First cut: distance between two points ( = distance between the end points of the two vectors) Euclidean distance? Euclidean distance is a bad idea because Euclidean distance is large for vectors of different lengths.

13 Sec. 6.3 Why distance is a bad idea The Euclidean distance between q and d 2 is large even though the distribution of terms in the query q and the distribution of terms in the document d 2 are very similar.

14 Sec. 6.3 Use angle instead of distance Thought experiment: take a document d and append it to itself. Call this document d. Semantically d and d have the same content The Euclidean distance between the two documents can be quite large The angle between the two documents is 0, corresponding to maximal similarity. Key idea: Rank documents according to angle with query.

15 Sec. 6.3 From angles to cosines The following two notions are equivalent. Rank documents in decreasing order of the angle between query and document Rank documents in increasing order of cosine(query,document) Cosine is a monotonically decreasing function for the interval [0 o, 180 o ]

16 Sec. 6.3 From angles to cosines But how and why should we be computing cosines?

17 Sec. 6.3 Length normalization A vector can be (length-) normalized by dividing each of its components by its length for this we use the L 2 norm: 2 x x 2 Dividing a vector by its L 2 norm makes it a unit (length) vector (on surface of unit hypersphere) Effect on the two documents d and d (d appended to itself) from earlier slide: they have identical vectors after length-normalization. Long and short documents now have comparable weights i i

18 cosine(query,document) V i i V i i V i i i d q q d d d q q d q d q d q ), cos( Dot product Unit vectors q i is the tf-idf weight of term i in the query d i is the tf-idf weight of term i in the document cos(q,d) is the cosine similarity of q and d or, equivalently, the cosine of the angle between q and d. Sec. 6.3

19 Similarity Measures Compared ), min( D Q D Q D Q D Q D Q D Q D Q D Q D Q Simple matching (coordination level match) Dice s Coefficient Jaccard s Coefficient Cosine Coefficient (what we studied) Overlap Coefficient

20 Summary vector space ranking Represent the query as a weighted tf-idf vector Represent each document as a weighted tf-idf vector Compute the cosine similarity score for the query vector and each document vector Rank documents with respect to the query by score Return the top K (e.g., K = 10) to the user

21 Evaluating ranked results: Mean Avg Precision 1 R 2 N 3 N 4 R 5 R 6 N 7 R 8 N 9 N 10 N Assume 10 rel docs in collection

22 Sec. 8.4 Common evaluation measure Mean average precision (MAP) AP: Average of the precision value obtained for the top k documents, each time a relevant doc is retrieved Avoids interpolation, use of fixed recall levels Does weight most accuracy of top returned results MAP for set of queries is arithmetic average of APs Macro-averaging: each query counts equally 29

23 Problems Synonyms: separate words that have the same meaning. E.g. car & automobile They tend to reduce recall Polysems: words with multiple meanings E.g. Java They tend to reduce precision The problem is more general: there is a disconnect between topics and words

24 a more appropriate model should consider some conceptual dimensions instead of words. (Gardenfors)

25 Latent Semantic Analysis (LSA) LSA aims to discover something about the meaning behind the words; about the topics in the documents. What is the difference between topics and words? Words are observable Topics are not. They are latent. How to find out topics from the words in an automatic way? We can imagine them as a compression of words A combination of words Try to formalise this

26 Latent Semantic Analysis Singular Value Decomposition (SVD) A(m*n) = U(m*r) E(r*r) V(r*n) Keep only k eigen values from E A(m*n) = U(m*k) E(k*k) V(k*n) Convert terms and documents to points in k- dimensional space Low-rank approximation

27 Latent Semantic Analysis Singular Value Decomposition {A}={U}{S}{V} T Dimension Reduction {~A}~={~U}{~S}{~V} T

28 Latent Semantic Analysis LSA puts documents together even if they don t have common words if The docs share frequently co-occurring terms Disadvantages: Statistical foundation is missing PLSA addresses this concern!

29 PLSA Latent Variable model for general co-occurrence data Associate each observation (w,d) with a class variable z Є Z{z_1,,z_K} Generative Model Select a doc with probability P(d) Pick a latent class z with probability P(z d) Generate a word w with probability p(w z) d z w N d D

30 PLSA To get the joint probability model

31 Model fitting with EM We have the equation for log-likelihood function from the aspect model, and we need to maximize it. Expectation Maximization (EM) is used for this purpose

32 EM Steps E-Step Expectation step where expectation of the likelihood function is calculated with the current parameter values M-Step Update the parameters with the calculated posterior probabilities Find the parameters that maximizes the likelihood function

33 E Step It is the probability that a word w occurring in a document d, is explained by aspect z (based on some calculations)

34 M Step All these equations use p(z d,w) calculated in E Step Converges to local maximum of the likelihood function

35

36 The performance of a retrieval system based on this model (PLSI) was found superior to that of both the vector space based similarity (cos) and a non-probabilistic latent semantic indexing (LSI) method. (We skip details here.) From Th. Hofmann, 2000

37 Comparing PLSA and LSA LSA and PLSA perform dimensionality reduction In LSA, by keeping only K singular values In PLSA, by having K aspects Comparison to SVD U Matrix related to P(d z) (doc to aspect) V Matrix related to P(z w) (aspect to term) E Matrix related to P(z) (aspect strength) The main difference is the way the approximation is done PLSA generates a model (aspect model) and maximizes its predictive power Selecting the proper value of K is heuristic in LSA Model selection in statistics can determine optimal K in PLSA

38 Latent Dirichlet Allocation Generative Model Per-document topic proportions Per-word topic assignment Observed word Per-topic word proportions α θ d z w β φ k N d K D

39 Joint Distribution φ i β φ 1:K 49

40 α = 1 50

41 α = 10 51

42 α =

43 α = 1 53

44 α =

45 α =

46 LDA vs. PLSA Explicit modeling of sparsity LDA was a rage Lots of tools Often gives similar results 56